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Theorem ialgrlem1st 12074
Description: Lemma for ialgr0 12076. Expressing algrflemg 6255 in a form suitable for theorems such as seq3-1 10491 or seqf 10492. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypothesis
Ref Expression
ialgrlem1st.f (𝜑𝐹:𝑆𝑆)
Assertion
Ref Expression
ialgrlem1st ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

Proof of Theorem ialgrlem1st
StepHypRef Expression
1 algrflemg 6255 . . 3 ((𝑥𝑆𝑦𝑆) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
21adantl 277 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
3 ialgrlem1st.f . . . 4 (𝜑𝐹:𝑆𝑆)
43adantr 276 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝐹:𝑆𝑆)
5 simprl 529 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
64, 5ffvelcdmd 5673 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹𝑥) ∈ 𝑆)
72, 6eqeltrd 2266 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  ccom 4648  wf 5231  cfv 5235  (class class class)co 5896  1st c1st 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243  df-ov 5899  df-1st 6165
This theorem is referenced by:  ialgr0  12076  algrp1  12078
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